Early on, properties related with arrival times of Poisson processes have been intensively
studied and now we can find comprehensive theories in textbooks
e.g. [1]}, [2]}, [3]}, [4]}. Among them the order statistics
property is one of the most important properties, which characterizes
the Poisson process and which is satisfied
only with the mixed Poisson process. See e.g. [1]} or [6]}, both of
which treat general (non-diffuse) mean measures.
However, although the Cox process is a
direct generalization of the mixed Poisson process, few attempts have
been made to arrival times of the Cox process.
One reason is that we have to treat random intensities which yield not
a little complexity in the analysis.
